Associative algebra deformations of the Connes-Moscovici's Hopf algebra $\mathcal{H}_1$

TL;DR

This paper computes the second Hochschild cohomology of Connes-Moscovici's Hopf algebra, revealing its deformation space and establishing the uniqueness of its formal deformation via Rankin-Cohen brackets.

Contribution

It provides the first explicit calculation of the Hochschild cohomology for $ ext{H}_1$, demonstrating the uniqueness of its associative algebra deformation.

Findings

$HH^2( ext{H}_1)$ is one-dimensional

The deformation using Rankin-Cohen brackets is unique up to equivalence

Explicit computation of Hochschild cohomology for $ ext{H}_1$

Abstract

We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence.